probabilistic construction of t-designs over finite...
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Probabilistic construction of t-designs over finite fields
Shachar Lovett (UCSD)
Based on joint works with Arman Fazeli(UCSD), Greg Kuperberg (UC Davis), Ron Peled
(Tel Aviv) and Alex Vardy (UCSD)
Gent workshop, 2013
t-designs over finite fields
• Finite field Fq
• t-(n,k,;q) design is a collection of k-dim subspaces in Fq
n, called blocks, such that each t-dim subspace of Fq
n
is contained in exactly blocks
• Trivial design: all k-dim subspaces• Question: find nontrivial designs
t-designs over finite fields
• t-designs over finite fields are an extension of the more standard notion of combinatorial t-designs, where subspaces are replaced by subsets
• Teirlinck’ 87: First construction of nontrivial combinatorial t-designs, for any t
• No analog theorem for designs over finite fields (constructions known only for t=1,2,3)
• This work: existence of nontrivial t-designs over finite fields, for any t– Proof by probabilistic argument, non constructive
Bigger picture
• t-designs over finite fields are an instance of “regular combinatorial objects”
• [Kuperberg-L-Peled’12]: General framework to prove existence of regular combinatorial objects by probabilistic techniques
• [Fazeli-L-Vardy’13]: Application to t-designs over finite fields
Regular combinatorial objects
• Example 1: Combinatorial t-designs
• Collection of k-subsets of {1,…,n}, called blocks, such that each t-subset of {1,…,n} is contained in exactly blocks
1
2
3 4 5
67
n=7,k=3,t=2,=1
Regular combinatorial objects
• Example 2: Orthogonal arrays
• Collection of vectors in [q]n, such that on any t coordinates, each one of the possible qt patterns appear exactly times
0 0 0
0 1 1
1 0 1
1 1 0
q=2,n=3,t=2,=1
Regular combinatorial objects
• Example 3: t-wise permutations
• Collection of permutations in Sn, such that for any indices i1,..,it and j1,…jt, the number of permutations mapping i1 to j1,i2 to j2,…,it to jt, is exactly
n=4,t=1,=1
1 2 3 42 3 4 13 4 1 24 1 2 3
Regular combinatorial objects
• Example 4: t-designs over finite fields
• Collection of k-dim subspaces of Fqn,
called blocks, such that each t-dim subspace of Fq
n is contained in exactly blocks
Regular combinatorial objects
• “highly symmetric” objects with many simultaneous conditions of exact counts
• Constructions known in special cases
• Existence cannot be exhibited by standard probabilistic techniques. Why?
Probabilistic constructions
• Consider, say, the problem of t-designs over finite fields
• If we choose randomly a small collection of k-dim subspaces (blocks), than any t-dim subspace will be in approximatelythe same number of blocks
• Approximately, but not exactly
KLP Framework
• Theorem [Kuperberg-L-Peled’12]: If the objects satisfy certain – symmetric properties,
– coding-theoretic properties, and
– divisibility properties,
then the probability that a random construction works is positive (but tiny)
• Hence, the required objects exist!
t-designs over finite fields
• [Fazeli-L-Vardy’13]
• Application of KLP framework
• Theorem: t-(n,k,;q) designs over a finite field F exist for any choice of Fq, t, k>12(t+1); and n large enough (n>>kt suffices)
• But, we don’t know how to find them efficiently…
Matrix averaging problem
• Let M be an integer matrix, with rows set R and columns set C– row(r) ZC
• We want to find a small subset S of rowswhose average equals the average of all the rows
1
|𝑆|
𝑟∈𝑆
𝑟𝑜𝑤 𝑟 =1
|𝑅|
𝑟∈𝑅
𝑟𝑜𝑤 𝑟
Matrix averaging problem
• For example, if: R = all k-dim subspacesC = all t-dim subspacesM = incidence matrix
• A subset S of rows for which
1
|𝑆| 𝑟∈𝑆 𝑟𝑜𝑤 𝑟 =
1
|𝑅| 𝑟∈𝑅 𝑟𝑜𝑤 𝑟
is exactly a t-design
010010100110010110000000101110
…
0010110100
k-d
im
subsp
aces
t-dim subspaces
Matrix averaging problem
• Can we hope that in general, in any 0-1 matrix, there are few rows whose average is the same as the average of all the rows?
• NO.• There are 0-1 matrices with |C|=n, |R|~nn/2
with no such subsets of rows [Alon-Vu]
• We, on the other hand, would like to have a subset of poly(n) rows
KLP theorem
• Theorem: If matrix M satisfies certain – symmetric properties,
– coding-theoretic properties, and
– divisibility properties,
then there is a small set of rows S such that
• Small = polynomial in |C|, other parameters
1
|𝑆| 𝑟∈𝑆
𝑟𝑜𝑤 𝑟 =1
|𝑅| 𝑟∈𝑅
𝑟𝑜𝑤 𝑟
KLP framework (1)
• Condition 1: all the elements in the matrix are small integers
• Trivially true for incidence matrices
010010100110010110000000101110
…
0010110100
010010100110010110000000101110
…
0010110100
KLP framework (2)
• V = subspace of QR spanned by columns
• Condition 2: constant vector in V
• For t-designs over finite fields, holds because sum of columns is a constant vector (#t-dim subsp. in a k-dim subsp.)
010010100110010110000000101110
…
0010110100
KLP framework (3)
• V = subspace of QR spanned by columns
• Symmetry group of V = group of permutations of rows which preserve V
• Condition 3: Symmetry group of V is transitive– e.g. for any pair of rows r1,r2 there is a
symmetry of V mapping r1 to r2
KLP framework (3)
• Example: t-designs over finite fields
• Rows = k-dim subsp., Cols = t-dim subsp.• V = subspace of QR spanned by columns
• GL(Fq,n) acts on rows and columns, preserve the incidence matrix. Hence, GL(Fq,n) < Sym(V)
• Action of GL(Fq,n) on R is transitive (can map any k-dim subspace to any k-dim subspace)
010010100110010110000000101110
…
0010110100
010010100110010110000000101110
…
0010110100
KLP framework (4)
• V = subspace of QR spanned by columns
• V = orthogonal subspace (in QR)
• Condition 4: V is spanned by short integer vectors
• Usually the hardest condition to verify
010010100110010110000000101110
…
0010110100
KLP framework (5)
• Condition 5: Divisibility. There exist a small integer c such that
expressible as integer combination of rows
• Necessary if we hope to get small S,
𝑐
|𝑅| 𝑟∈𝑅
𝑟𝑜𝑤 𝑟
1
|𝑆| 𝑟∈𝑆
𝑟𝑜𝑤 𝑟 =1
|𝑅| 𝑟∈𝑅
𝑟𝑜𝑤 𝑟
KLP theorem
• Theorem: If matrix M satisfies certain – symmetric properties,
– coding-theoretic properties, and
– divisibility properties,
then there is a small set of rows S such that
• Small = polynomial in |C|, other parameters
1
|𝑆| 𝑟∈𝑆
𝑟𝑜𝑤 𝑟 =1
|𝑅| 𝑟∈𝑅
𝑟𝑜𝑤 𝑟
Proof idea
• S = random small set of rows
• Analyze the probability that
• If the conditions hold, can approximate probability up to 1+o(1) by an appropriate Gaussian process restricted to a lattice
• Proof utilizes new connections between Fourier analysis, coding theory and local central limit theorems
1
|𝑆| 𝑟∈𝑆
𝑟𝑜𝑤 𝑟 =1
|𝑅| 𝑟∈𝑅
𝑟𝑜𝑤 𝑟
Summary
• New probabilistic technique
• Can prove existence of regular combinatorial structures
• Application: t-designs over finite fields
Open problems (1)
• Algorithmic: Can prove existence, but we don’t know how to find the objects efficiently
• For other probabilistic techniques for “rare events” this was accomplished – Lovász Local Lemma [Moser, Moser-Tardos,…]– Spencer’s “six standard deviations suffice”
[Bansal, L-Meka]
• So, I am hopeful…
Open problems (2)
• Other applications
• Large sets (e.g. partitions)
• Sparse systems (Steiner systems, Hadamard matrices)
Open problems (3)
• Perfect pseudo-randomness in group theory
• Conjecture: for any group G acting transitively on a set X, there is a small subset SG such that S acts uniformly on X,
|{gS: g(x)=y}|=|S|/|X| x,yX
• Proved for G=Sn, S=all k-sets• Open: G=GL(n,F); S=k-dim Grasmannian